Plurality Voting with Truth-biased Agents - PowerPoint Presentation

Slide1

Plurality Voting with Truth-biased Agents

Vangelis Markakis

Joint work with:Svetlana Obraztsova, David R. M. Thompson

Athens University

of

Economics and Business (AUEB)

Dept. of InformaticsSlide2

Talk OutlineElections – Plurality Voting

Game-theoretic approaches in votingTruth-bias: towards more realistic modelsComplexity and characterization results

Pure Nash EquilibriaStrong Nash EquilibriaConclusions2Slide3

SetupElection

s:a set of candidates C = {c1

, c2,…,cm}a set of voters V = {1, ..., n}for each voter i, a preference order aieach ai is a total order over Ca = (a1, …, an): truthful profilea voting rule F:given

a ballot vector

b

= (b

1

, b

2, …,bn), F(b) = election outcome we consider single-winner elections

3Slide4

Setup

For this talk, F = Plurality The winner is the candidate with the maximum number of votes who ranked him first

Lexicographic tie-breaking: w.r.t. an a priori given orderAmong the most well-studied voting rules in the literature4Slide5

Strategic Aspects of Voting

Gibbard-Satterthwaite theorem5

For |C|>2, and for any non-dictatorial voting rule, there exist preference profiles where voters have incentives to vote non-truthfully Slide6

Strategic Aspects of VotingBeyond Gibbard-Satterthwaite

:Complexity of manipulationManipulation by coalitionsEquilibrium analysis (view the election as a game among selfish voters)Study properties of Nash Equilibria

or other equilibrium concepts6Slide7

A Basic Game-theoretic Model

Players = votersStrategies = all possible votesWe assume all voters will cast a vote

Utilities: consistent with the truthful preference order of each voter We are interested in (pure) Nash Equilibria (NE)[Initiated by Farquharson ’69] 7Slide8

Undesirable NE under Plurality

8

5 voters deciding on getting a pet

Truthful profileSlide9

Undesirable NE under Plurality

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It is a NE for all voters to vote their least preferred candidate!

Problems with most voting rules under the basic model:

Multitude of Nash

equilibria

Many of them unlikely to occur in practice

5 voters deciding on getting a pet

Truthful profileSlide10

Can we eliminate bad NE?

10Some ideas:

Strong NE: No coalition has a profitable deviation [Messner, Polborn ’04, Sertel, Sanver ’04]Drawback: too strong requirement, in most cases they do not exist Voting with abstentions (lazy voters) [Desmedt, Elkind ’10]Small cost associated with participating in votingDrawback: it eliminates some equilibria, but there can still exist NE where the winner is undesirable by most players Slide11

Truth-biased Voters

Truth-bias refinement: extra utility gain (by ε>0) when telling the truth

if a voter cannot change the outcome, he strictly prefers to tell the truthε is small enough so that voters still have an incentive to manipulate when they are pivotal11More formally:Let c = F(b), for a ballot vector b = (b1, b2, …,bn)Payoff for voter i is: ui(c) + ε, if i

voted truthfully

u

i

(c

),

otherwiseSlide12

Truth-biased Voters

T

he snake can no longer be elected under truth-bias

Each voter would prefer to withdraw support for the snake and vote truthfully

12Slide13

Truth-biased Voters

Truth bias achieves a significant elimination of “bad” equilibria

Proposed in [Dutta, Laslier ’10] and [Meir, Polukarov, Rosenschein, Jennings ’10]Experimental evaluation: [Thompson, Lev, Leyton-Brown, Rosenschein ’13]Drawback: There are games with no NEBut the experiments reveal that most games still possess a NE (>95% of the instances)

Good social welfare properties (“undesirable” candidates not elected at an equilibrium)

L

ittle theoretical analysis so far

Questions of interest:

Characterization of NE

Complexity of deciding existence or computing NE

13Slide14

Complexity Issues

Theorem: Given a score s, a candidate c

j and a profile a, it is NP-hard to decide if there exists a NE, where cj is the winner with score s.Proof: Reduction from MAX-INTERSECT [Clifford, Poppa ’11]ground set E, k set systems, where each set system is a collection of m subsets of E, a parameter q. ``Yes''-instance: there exists 1 set from every set system s.t. their intersection consists of ≥ q elements.14Slide15

Complexity Issues

15Hence:

Characterization not expected to be “easy”But we can still identify some properties that illustrate the differences with the basic modelSlide16

An Example

16Truthful profile a

= (a1,…,a6) with 3 candidatesTie-breaking: c1 > c2 > c3c1 = F(a), but a is not a NE 123456c1c

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



Non-truthful profile

b

c

2

= F(

b

), and

b

is a NE



Non-truthful profile

b’

c

2

= F(

b’

), but

b’

is not a NE

“too many” non-truthful votes for c

2

Slide17

Warmup: Stability of the truthful profile

Theorem: Let ci

= F(a), be the winner of the truthful profileThe only possible NE with ci as the winner is a itselfWe can characterize (and check in poly-time) the profiles where a is a NE17Proof: Simply use the definition of truth-bias. If  NE b ≠ a, true supporters of ci would strictly prefer to vote truthfully.

n

on-supporters

of

c

i

also do not gain by lying in b, hence they prefer to be truthful as well

(2) The possible threats to c

i

in

a

are only from candidates who have equal score or are behind by one vote. Both are checkable in poly-timeSlide18

Non-truthful NEGoal:

Given a candidate cj, a score

s, the truthful profile a,Identify how can a non-truthful NE b arise, with cj = F(b), and score(cj, b) = s18Slide19

Key Properties under Truth-bias

Lemma 1: If a non-truthful profile is a NE then all liars in this profile vote for the current winner (not true for the basic model)

19Definition: A threshold candidate w.r.t. a given profile b, is a candidate who would win the election if he had 1 additional voteLemma 2: If a non-truthful profile b is a NE, then there always exists ≥ 1 threshold candidate (not necessarily the truthful winner)such candidates have the same supporters in b as in aIntuition: In any non-truthful NE, the winner should have “just enough” votes to win, otherwise there are non-pivotal liars Slide20

Conditions for existence of NE

nj := score of

cj in the truthful profile ac* := winner in a, n* = score(c*, a)Claim: If such a NE exists, then nj ≤ s ≤ n* + 1, Lower bound: cj cannot lose supporters (Lemma 1)Upper bound: in worst-case, c* is the threshold candidate

20Slide21

Conditions for existence of NEVotes in favor of

cj in b

:nj truthful voterss – nj liars Q: Where do the extra s – nj voters come from?21Slide22

Conditions for existence of NE

Eventually we need to argue about candidates with:nk ≥ snk

= s-1 nk = s-2All these may have to lose some supporters in b towards cjExcept those who are threshold candidates (by Lemma 2)Notation:T: inclusion-maximal s-eligible threshold set i.e., the set of threshold candidates if such a NE existswe can easily determine T, given cj, s, and aM≥r: the set of candidates whose score is ≥ r in a

22Slide23

Conditions for existence of NE

Main results:Full characterization for having a NE b with:cj

= F(b)Score of cj = s Threshold candidates w.r.t. b = T’, for a given T’Implications:Identification of tractable cases for deciding existenceNecessary or sufficient conditions for the range of s – nj23Slide24

Conditions for existence of NECase 1

: All candidates in T have score s-1 in

a. 24We have a“no"-instance if:“yes”-instance if

: Slide25

Conditions for existence of NE25

0

Possible values for s - njNo NE b with cj = F(b)



NE

b

with

c

j = F(b)

NP-hard to decide

We can obtain

much bett

er

refinements of these intervals

Details in the paper…Slide26

Conditions for existence of NE26

Case 2:

There exists a candidate in T whose score in a is s.

W

e

have a

“no"-instance if

:

“yes”-instance if: Slide27

Strong Nash Equilibria

Definition: A profile b is a strong NE if there is no coalitional deviation that makes all its members better off We have obtained analogous characterizations for the existence of strong NE

Corollary 1: We can decide in polynomial time if a strong NE exists with cj as the winnerCorollary 2: If there exists a strong NE with cj = F(b), then cj is a Condorcet winnerOverall: too strong a concept, often does not exist27Slide28

Conclusions and Current/Future Work

Truth bias: a simple yet powerful idea for equilibrium refinement Iterative voting: study NE reachable by iterative best/better response updates

Unlike basic model, we cannot guarantee convergence for best-response updates [Rabinovich, Obraztsova, Lev, Markakis, Rosenschein ’14]Comparisons with other refinement models (e.g. lazy voters) or with using other tie-breaking rules? [Elkind, Markakis, Obraztsova, Skowron ’14]28Slide29

Conditions for existence of NECase 1

: All candidates in T have score s-1 in

a. Then we have a“no"-instance if: s - nj ≤ ∑ nk – (s-3)|M≥s-1\T|“yes”-instance if: s - nj ≥ ∑ nk – (s-3)|M≥s-2\T|ckϵM≥s-1\T

c

k

ϵ

M

≥s-2

\T29Slide30

Key Properties under Truth-bias

Lemma: If a non-truthful profile is a NE then all liars in this profile vote for the current winner (not true for the basic model)Definition:

A threshold candidate for a given set of votes is a candidate who would win the election if he had 1 additional voteLemma: If a non-truthful profile is a NE, then there always exists ≥ 1 threshold candidate

Tie-breaking:

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˃

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One more example

Tie-breaking:

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One more example

Tie-breaking:

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